Quiver Schur algebras for the linear quiver I

Jun Hu and Andrew Mathas

Abstract

We define a graded quasi-hereditary covering for the cyclotomic quiver Hecke algebras ${\mathcal{R}}_n^{\mathrm{\Lambda }}$ of type $A$ when $e=0$ (the linear quiver) or $e\ge n$. We show that these algebras are quasi-hereditary graded cellular algebras by giving explicit homogeneous bases for them. When $e=0$ we show that the KLR grading on the quiver Hecke algebras is compatible with the gradings on parabolic category ${\mathcal{O}}_n^{\mathrm{\Lambda }}$ previously introduced in the works of Beilinson, Ginzburg and Soergel and Backelin. As a consequence, we show that when $e=0$ our graded Schur algebras are Koszul over field of characteristic zero. Finally, we give an LLT-like algorithm for computing the graded decomposition numbers of the quiver Schur algebras in characteristic zero when $e=0$.

Keywords: Cyclotomic Hecke algebras, Schur algebras, quasi-hereditary and graded cellular algebras, Khovanov-Lauda-Rouquier algebras.

AMS Subject Classification: Primary 20C08; secondary 20C30, 05E10.